At convergence, the interface data at \(\partial \Omega _{1,2}\)  will satisfy the higher-order continuity prescriptions specified by the user, thereby guaranteeing full regularity of \(C^{p-1}\) . The illustration in Fig. (REF ), and the methodology description in this section can be generalized and extended to arbitrary dimensions in the tensor-product setting (with the parametric domain represented by a \(d-\) dimensional hypercube) as shown in Fig. (REF ) for 2D. Using knot insertion and removal strategies based on deCasteljau subdivision procedures [1]}, individual subdomains can also be adapted to resolve fast varying solutions and to reduce decoded error to be within user-specified tolerances [2]}. While adaptivity has not been fully explored in the current work, enabling variable resolutions in different dimensions is a natural extension of the work that will still preserve high-order continuity.
The implementation of the presented approach with domain decomposition strategies, combined with overlapping RAS scheme yields a scalable scheme that will be demonstrated to be suitable for tackling large-scale data analysis problems.
